The common tangents to the circle x 2 + y 2 = 2 and the parabola y 2 = 8 x touch the circle at the points P , Q and the parabola at the points R , S . Then the area of the quadrilateral P Q R S is [2014]

Question

The common tangents to the circle $x^{2} + y^{2} = 2$ and the parabola $y^{2} = 8 x$ touch the circle at the points $P, Q$ and the parabola at the points $R, S$ . Then the area of the quadrilateral $P Q R S$ is [2014]

Smart Achievers · Accepted Answer

(4)

$Let the tangent to y^{2} = 8 x be y = m x + \frac{2}{m}$

If it is common tangent to parabola and circle $x^{2} + y^{2} = 2$ , then distance of the tangent from the centre of the circle is equal to radius of the circle

$∴$ $| \frac{\frac{2}{m}}{\sqrt{m^{2} + 1}} |$ $= \sqrt{2} \Rightarrow \frac{4}{m^{2} (1 + m^{2})} = 2$

$\Rightarrow m^{4} + m^{2} - 2 = 0 \Rightarrow (m^{2} + 2) (m^{2} - 1) = 0 \Rightarrow m = 1 or - 1$

$∴$ $Required tangents are y = x + 2 and y = - x - 2$

$Their common point is (- 2, 0)$

$∴$ $Tangents are drawn from (- 2, 0)$

$∴$ $Chord of contact P Q to circle is$ $x . (- 2) + y . (0) = 2 \Rightarrow x = - 1$

$and chord of contact R S to parabola is$ $y (0) = 4 (x - 2) \Rightarrow x = 2$

$Hence coordinates of P and Q are (- 1, 1) and (- 1, - 1) respectively.$

$Also coordinates of R and S are (2, - 4) and (2, 4) respectively.$

$∴$ $Area of trapezium P Q R S = \frac{1}{2} (2 + 8) \times 3 = 15$

Solution

Q.
The common tangents to the circle $x^{2} + y^{2} = 2$ and the parabola $y^{2} = 8 x$ touch the circle at the points $P, Q$ and the parabola at the points $R, S$ . Then the area of the quadrilateral $P Q R S$ is [2014]

1 3

2 6

3 9

4 15

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Solution

Q. The common tangents to the circle x2+y2=2 and the parabola y2=8x touch the circle at the points P,Q and the parabola at the points R,S. Then the area of the quadrilateral PQRS is [2014]

1 3

2 6

3 9

4 15

Want Us to Email you About Special Offers & Updates?

Q.
The common tangents to the circle $x^{2} + y^{2} = 2$ and the parabola $y^{2} = 8 x$ touch the circle at the points $P, Q$ and the parabola at the points $R, S$ . Then the area of the quadrilateral $P Q R S$ is [2014]